Abstract
This study had succeeded in producing a new graphical representation of James abacus called nested chain abacus. Nested chain abacus provides a unique mathematical expression to encode each tile (image) using a partition theory where each form or shape of tile will be associated with exactly one partition.Furthermore, an algorithm of nested chain abacus movement will be constructed, which can be applied in tiling theory.
Highlights
About 15 years after James abacus was introducedJames abacus diagram is a graphical representation of a special type of non-increasing sequenceμ = (μ1, μ2,...,μb)called the partition of twhere b is the number of partition parts and t is any positive integer
Theorem 14: Let amj be an element in a rectangle chain in the nested chain abacus with e columns, r rows and c chains represented by matrix Ar×e
Corollary 15:Let amj be an element in a rectangle chain in the nested chain abacus with 2 columns, r rows and c chains represented by matrix Ar×2
Summary
1), 1), d∈Z c ∈ Z, which is in anticlockwise direction for all bead positions in chain i in the nested chain abacus with e columns and r rows where0 ⩽ m ⩽ r − 1and The bead positions in the rectangle chain i are located in columns {i, e − i + 1} and rows Let amj be an initial position in rectangle chain i for the nested chain abacus with e columns, r rows and c chain.
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